Chapter-2 Literature Review

Transcription

1 Chapter-2 Literature Review ii

2 CHAPTER - 2 LITERATURE REVIEW Literature review is divided into two parts; Literature review of load flow analysis and capacitor allocation techniques. 2.1 LITERATURE REVIEW FOR LOAD FLOW ANALYSIS Load-flow analysis of distribution system is an important area of activity as it is the final link between distribution systems and consumers. Load flow is the fundamental numerical algorithm for power system analysis. A few methods have been reported in the literature survey for load-flow analysis of distribution system. Tinney et. al. [16] developed the classical Newton based power flow solution method. Stott and Alsac [17] made the fast decoupled Newton method but its convergence performance is slow for the distribution systems due to their high R/X ratio. For this reason, several non-newton type of methods have been developed. Iwamoto and Tamura [18] modified the conventional methods to solve the distribution networks but this method was complex and quite time consuming. Method for solving ill-conditioned power systems was proposed by Tripathy et. al. [19]. This method showed voltage convergence but could not be used efficiently for optimal powerflow calculations. Kersting [20] developed a technique for solving the loadflow problem in radial distribution networks based on ladder network theory in the iterative routine. This load flow solution was complicated and had many assumptions for a typical distribution system but it was not designed efficiently to solve meshed networks. The ladder based technique 41

3 proposed by Stevens et. al. [21] was very fast but did not guarantee convergence. Shirmohammadi et. al. [22] presented a new compensation based power flow method for the solution of weakly meshed distribution networks. This technique solved radial distribution networks with the help of direct voltage application of Kirchhoff s laws and presented a branchnumbering scheme to enhance numerical performance of the method. This method undoubtedly was more efficient than the Newton-Raphson power flow technique when used for solving radial and weakly meshed distribution and transmission networks but needed a rigorous data preparation. Renato [23] proposed a method for obtaining load-flow solution of radial distribution networks computing the electrical equivalent for each node and summing all the loads of the network fed through the node including losses. Then starting from the source node, voltage of each receiving-end node was computed but the angle of the voltage was not considered in this method. Baran et.al. [24] presented a forward method in which starting from the sending end voltage the system was converged. Voltage drop and the information on system structure had been considered in the forward sweep. The voltage- sensitive load current could be included in the system model. However, this method still had disadvantages. Oriented from ladder network concepts, the branch flow equations were solved by a Newton- Raphson approach which made this method complex and costly. The influence of the load distribution would cause slow convergence. Luo et.al. [25] presented a compensation method for weakly meshed networks. This method started from a network structure analysis to find the interconnection points. Then it broke those interconnection points using the compensation method so that the meshed system structure could be changed to simple tree-type radial system. This method was also suitable for the system with 42

4 multiple voltage control buses. This method had some disadvantages such as the structure analysis of the system was complex. Sometimes a heuristic method was used to decide the break points. So the adaptability of this method was not good enough. The other disadvantage is that load conditions at the break points have great influence on the power flow solutions and heavy loading loops and weak power sources (dispersed generation) in meshed systems may cause difficulties for the speed and accuracy of convergence. Goswami and Basu [26] presented an approximate method for solving radial and meshed distribution networks. This method had the advantages of a guaranteed accurate solution for a realistic distribution system with composite loads and without any problem of convergence. But the disadvantages were difficulty in numbering the nodes and branches and no node in the network was the junction of more than three branches. Chiang [27] presented three different algorithms for solving radial distribution networks. The decoupled and fast decoupled distribution load-flow algorithms presented by Chiang were similar to that of Baran and Wu. However, the very fast decoupled distribution load-flow presented by Chiang was very attractive because it did not require any construction of Jacobian matrix and factorization. Jasmon and Lee [28] presented a load-flow technique for every branch, which gives pair of quadratic equations relating power flows at both ends with the voltage magnitude at the sending end in connection with the voltage stability analysis of radial networks. Das et.al. [29] presented a load-flow method for the convergence of power with the help of coding at the lateral and sub lateral nodes which increased complexity of computation of large system. This method worked only for sequential branch and node numbering scheme. Here, the voltage was calculated at each receiving end node using 43

5 forward sweep and the initial guess of power loss was taken as zero to solve radial distribution networks (RDN). It solved the simple algebraic recursive expression of voltage magnitude and all the data were stored in vector form. The algorithm used the basic principle of circuit theory. This method had the advantage that all data could be stored in vector forms and thus saving an enormous amount of computer memory. Kalam and Das [30] suggested a straight forward solution which doesn t depend on the phase angle and this simplifies the formulation of the problem. In distribution system, the voltage angle is not so important because the variation of voltage angle from the substation to tail of distribution feeder is only few degrees. Haque [31] proposed a new approach for meshed networks with more than one feeding node. This method first converted the multiple-source mesh network into an equivalent single-source radial type network by setting dummy nodes for the break points at distributed generators and loop connecting points. Then the traditional ladder network method could be applied for the equivalent radial system. Following each of the iterations of the equivalent radial system, the power injected at the breaker points must be updated by an additional calculation through a reduced order impedance matrix. A phase decoupled load-flow method was proposed by Lin and Teng [32] in which fast convergence was ensured by means of the Newton-Raphson algorithm using branch currents as state variables. Keeping into account the mutual relations between each phase s parameters, some simplifications were introduced so as to obtain three sub- Jacobian constant matrices in the main Jacobian matrix. Nguyen [33] presented a new algorithm which is based on the extension of Newton- Raphson method and its Jacobian in complex form was used for the solution of three phase power flow analysis of both transmission or 44

6 distribution systems under unsymmetrical operating conditions. This method gave the solutions in whole phasor format which made it suitable for applications in voltage quality analysis and power quality improvement. Jacobian matrix increased the memory requirement of this method. Expositos and Ramos [34] presented a method to solve the power flow problem in radial networks. In this formulation, the load-flow equations were written in terms of new variables resulting in a set of 3N equations (2N linear plus N quadratic) for a network with N+l buses. Here, a computationally efficient solution scheme based on the Newton-Raphson method was presented and some simplifications were also discussed. Lin et.al. [35] proposed an exact three-phase fast decoupled power flow solution for radial distribution system. This method used traditional Newton-Raphson algorithm in a rectangular coordinate system and the Jacobian matrix of the presented method could be decoupled both on phases as well as on real and imaginary parts. In addition, the memory requirement of the traditional fast decoupled load-flow could be reduced to only one- sixth and the need of the complicated mutual coupling terms could be avoided. Here, it was even possible to solve the distribution system with line conductance only. Hence it is an exact three-phase distribution load-flow program which could be executed with minimum data preparation and substantially off-loading the burden of distribution engineers. Ghosh and Das [36] presented a load- flow method for solving radial distribution networks by evaluating only a simple algebraic expression of receiving end voltages. In this method, the authors assumed an initial flat voltage for all nodes. Then, by numbering the nodes beyond each branch, they calculated the loads and charging currents, followed by the branch currents. The modified nodal voltages and losses were also 45

7 calculated. Evaluating the difference between new and previous voltage values and then comparing it with an accepted tolerance for verifying the convergence. This method was simple and had good and fast convergence and could be used for composite load modeling, if the composition of the loads was known. The main drawback of this method was that it stored nodes beyond each branch and calculated current for each branch by adding load currents of nodes beyond the respective branch. Augugliaro et.al. [37] presented a fast converging method for the load-flow analysis of radial distribution networks. This method was based on an iterative algorithm with some special procedures to increase the convergence speed. The bus voltages were considered as state variables and used a simple matrix representation for the network topology and branch current flow management. Aravindhababu et.al. [38] presented a simple and efficient branch-to-node matrix-based power flow (BNPF) for radial distribution systems but this method was unsuitable for its extension to optimal power flow for which the NR method seems to be more appropriate. In that method any presence of sub laterals complicated the matrix formation. Afsari et.al. [39] presented a load-flow method based on estimation of node voltage and it was also assumed that the loads of the nodes of lateral and their sub lateral were concentrated at the originating node of the feeder. Here it was tried to reduce the computation time only but the computation became very complex when the number of laterals and sub-laterals increased. Mekhamer et.al. [40] used the equations developed by Baran and Wu for each node of the feeder but used different procedure. In this method the load-flow problem was solved by considering the laterals as a concentrated load of the main feeder. Once the voltage of the main feeder calculated, the first node voltage of each lateral was put equal to the voltage 46

8 of the same node on the main feeder. The node voltages of the lateral were then calculated using Baran and Wu equations. The convergence criterion was made upon the active and reactive power fed through the terminal nodes of laterals and main feeder. Ranjan and Das [41] had presented a simple and efficient method to solve radial distribution network. This method solves the simple algebraic recursive expression of voltage magnitude and all the data were stored in vector form thus saving computer memory. The algorithm used the basic principle of circuit theory. Eminoglu and Hocaoglu [42] presented a simple and efficient method to solve the power low problem in radial distribution systems. This method took into account voltage dependency of static loads and line charging capacitance. The method was based on the forward and backward voltage updating by using polynomial voltage equation for each branch and backward ladder equation (Kirchhoff s Laws). The convergence ability and reliability of the method was compared with the Ratio-Flow method, which was based on classical forward backward ladder method, for different loading conditions, R/X ratios and different source voltage levels, under the wide range of exponents of loads. Satyanarayana et. al. [43] developed the loadflow equations from the A B C D parameter model of short transmission line and applied the method on the different load models. Golkar [44] proposed a method in which the matrix is not large and so the amount of computer memory used is much less as compared to conventional ones. Hence, the formulation is very simple and the method is very fast. In this formulation, the branches are represented by their single-line diagram and the loads are assumed to be balanced. In this method, the analysis begins by assuming initial values for the bus voltages. The currents taken by different buses are calculated starting from the end buses to the source. The source 47

9 bus current is updated and the branch currents are again calculated from the source to the end buses. The calculations are repeated until the difference between the losses calculated in 2 consecutive iterations becomes considerably low. Ghosh and Sherpa [45] presented a method for load-flow solution of radial distribution networks with minimum data preparation. Here the node and branch numbering need not to be sequential like other available methods. This method did not need sending-node; receiving-node and branch numbers if those were sequential. The presented method used the simple equation to compute the voltage magnitude and had the capability to handle composite load modeling. Here, the set of nodes of feeder, laterals and sub laterals were used. Most recently, Kumar and Arvindhababu [46] presented an approach of power flow with a view to obtain a reliable convergence in distribution systems. The trigonometric terms were eliminated in the node power expressions and thereby the resulting equations were linearised partially for obtaining better convergence. This method was simpler than existing approaches and solved iteratively similar to Newton-Raphson (NR) technique. Here, a novel approach [47] for formulating the power flow problem is based on the vector continuous Newton s method. This approach shows that there is a formal analogy between the Newton s method and a set of autonomous ordinary differential equations. The analogy suggests that any efficient numerical integration method can be used for solving the power flow problem. Nagaraju et.al. [48] proposed a method for load flow of radial distribution networks using sparse technique. But this method is only suitable for the sequentially numbered networks. If the network is not sequential numbered then manual work has to be done to make it sequentially numbered. Hamouda and Zehar [49] proposed an algorithm for 48

10 load-flow solution of radial distribution networks in which branch-to-node matrix is constructed and then inverse of this matrix is formed to get the node-to-branch network. From node-to-branch matrix a branch matrix is formed. All this procedure needs ample amount of computational effort and computer memory to store the matrices. Here, a new efficient method [50] is proposed for load-flow solution of radial distribution networks and simple transcendental equations are used to relate the sending-end voltage, receiving-end voltage and voltage drops in each branch of the distribution system. The effect of charging capacitance of the line has been incorporated in load-flow solution. A computer algorithm is developed in such a way that there is no need to adopt any sequential node numbering scheme for the solution of the networks. The angle of the receiving-end voltage is also computed along with the magnitude of the voltage. The voltage magnitude and angle are updated after each successive iteration and the voltage drops are then computed by using the new obtained values of voltage magnitude and angle. Hongbo et.al. [51] proposed a hybrid decoupled power flow method for balanced power distribution systems with distributed generation sources. This method formulates the power flow equations in active power and reactive power in decoupled form with polar coordinates. Second-order terms are included in the active power mismatch iteration, and constant Jacobian and Hessian matrices are used and a hybrid direct and indirect solution technique is used to achieve efficiency and robustness of the algorithm. Active power correction is calculated by means of a sparse lower triangular and upper triangular (LU) decomposition algorithm and using the restarted generalized minimal residual algorithm with an incomplete LU pre-conditioner. 49

11 2.2 LITERATURE REVIEW FOR CAPACITOR PLACEMENT TECHNIQUES The optimal capacitor placement is a complex optimization problem. Many techniques and algorithms have been proposed in this regards in past ANALYTICAL METHODS Analytical methods mainly used the calculus of capacitor placements to reduce losses and maximum savings. These methods were based on impractical assumptions like constant conductor size, uniform loading, non discrete capacitor sizes, equal capacitor sizes and constant capacitor locations. Due to these analytical methods, the famous two-thirds rule was established. According to two-thirds rule, a capacitor of rating equal to twothirds of the peak reactive demand should be installed at a position twothirds of the distance along the total feeder length for maximum loss reduction. The early analytical methods for capacitor placement are mainly developed by Neagle and Samson [52]. The problem is defined as determining the location and size of a given number of fixed type capacitors to minimize the power loss for a given load but the cost of capacitors and the changes in the node voltages were neglected. Further a voltage independent reactive current model was formulated and solved by Cook [53] & Bae [54]. Lee et al. [55] used the fixed and switched capacitors which were placed for optimizing the net monetary savings associated with the reduction of power and energy losses. Grainger et al. [56] proposed a voltage dependent methodology for shunt capacitor compensation of primary distribution feeders. Kaplan [57] proposed analytical method to optimize number, location and size of capacitors. Grainger et al. [58]-[61] formulated equivalent normalized feeders which 50

12 considered feeder sections of different conductor sizes and non- uniformly distributed loads NUMERICAL PROGRAMMING METHODS Numerical programming methods are iterative techniques which are used to maximize or minimize an objective function of decision variables. For optimal capacitor allocation, the losses or the savings function would be the objective function. Bus voltages, current, capacitor available size and number of capacitors may be the decision variables. The values of these decision variables must also satisfy a set of constraints. Duran [62] used discrete capacitors for a feeder with many sections of different wire sizes and concentrated loads and proposed a dynamic programming solution method. Grainger et al. [63] formulated the problem as a nonlinear programming problem by treating the capacitor sizes and the locations as continuous variables. Ponnavaiko and Rao [64] considered load growth as well as system capacity release and voltage rise at light load conditions and used a local optimization technique called the method of local variables by treating capacitor sizes as discrete variables. Grainger and Civanlar [65] combined capacitor placement and voltage regulator problems for a general distribution system and proposed a decoupled solution methodology. Baran and Wu [66] determined the optimal size of capacitors placed on the nodes of a radial distribution system and minimized the power losses for a given load by nonlinear programming using decomposition technique. Baldick and Wu [67] used integer quadratic programming to coordinate the optimal operation of capacitors and regulators in a distribution system. [68] proposed a mixed integer programming for the optimal location and sizing of static and switched shunt capacitors in radial distribution systems. 51

13 2.2.3 HEURISTIC METHODS Heuristics methods are developed through experience and judgment. The purpose of developing such heuristic techniques is to decrease the exhaustive search space, while keeping the end result of objective function at an approximate optimal value. The results produced by heuristics algorithms are not guaranteed to be optimal. Salam et.al [69] presents a heuristic strategy based on [70] with varying load to reduce system losses by identifying sensitive nodes at which capacitors should be placed. The capacitor size is the value that yields minimum system real losses without violation of voltage. The process is repeated for the next candidate node until no further losses reduction is achieved. [71] proposed a method of minimizing the loss associated with the reactive component of branch currents by placing optimal capacitors at proper locations. Once the capacitor locations are identified, the optimal capacitor size at each selected location for all capacitors is determined simultaneously, to avoid overcompensation at any location, through optimizing the loss-saving equations. Chis et al. [72] considered the varying load and optimized capacitor sizes based on maximizing the net economic savings from both energy and peak power loss reductions. Hamada et al. [73] introduced a new strategy for capacitor allocation handling the reduction in the section losses by adding a new voltage violation constraint. The new constraint has been the sectional ohmic losses in each branch of the feeder. 52

14 2.2.4 ARTIFICIAL INTELLIGENCE METHODS AI methods include genetic algorithms, artificial neural network, simulation annealing, fuzzy logic method, particle swarm optimization, ant algorithm. These methods for capacitor allocation are discussed below: A. Genetic Algorithm Genetic programming is based on the Darwinian principle of reproduction and survival of the fittest and analogous to naturally occurring genetic operations such as crossover and mutation [74]. H. Kim and S.K You [75] have used genetic algorithm for obtaining the optimum values of shunt capacitor bank. They have treated the capacitors as constant reactive power loads. S. Sundhararajan and Anil Pahwa [76] proposed an optimization method using genetic algorithm to determine the optimal selection of capacitors. To achieve high performance and high efficiency of the proposed algorithm, an improved adaptive genetic algorithm (IAGA) [77] is developed to optimize capacitor switching, and a simplified branch exchange algorithm is developed to find the optimal network structure for each genetic instance at each iteration of capacitor optimization algorithm. In [78], a nested procedure is proposed to solve the optimal capacitor placement problem for distribution networks. At the outer level, a reducedsize genetic algorithm is adopted which is aimed at maximizing the net profit associated with the investment on capacitor banks. At the inner level, power losses are minimized for the remaining loading conditions, taking into account the capacitor steps determined at the outer level. [79] proposed a genetic algorithm as search method to determine optimum value of injected reactive power while considering the effects of loads harmonic component on network. 53

15 B. Artificial Neural Network A two-stage artificial neural network is used to control the capacitors installed on a distribution system for a non-uniform load profile [80]. Gu et. al. [81] control both capacitor banks and voltage regulators using artificial neural network. The regulation of capacitor banks by parallel application of artificial neural network and genetic algorithm is proposed in [82]. The combined use of both methods of the artificial intelligence allows solving the reactive power and voltage control problem with higher level of its reliability and quality. C. Simulated Annealing Simulated annealing (SA) is based on annealing of materials which involves heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. The heat causes the atoms to become unstuck from their initial positions (a local minimum of the internal energy) and wander randomly through states of higher energy; the slow cooling gives them more chances of finding configurations with lower internal energy than the initial one. By analogy with this physical process, each step of the SA algorithm attempts to replace the current solution by a random solution. The new solution may then accepted with a probability which depends both on the difference between the corresponding function values and also on a global parameter T (called the temperature), that is gradually decreased during the process. Chiang et al. [83] used the optimization techniques based on simulated annealing (SA) to search the global optimum solution to the capacitor placement problem. This methodology is proposed to determine the locations where capacitors are to be installed, the types and sizes of capacitors to be installed, and the control settings of 54

16 these capacitors at different load levels. Ananthapadmanabha et al. [84] used SA to reduce the cost function which includes the energy and the capacitor installation cost. The practical aspects of capacitors, load constraints and operational constraints at different load levels are considered in [85] which are solved by a powerful simulated annealing approach. D. Fuzzy Logic Method This method is based on fuzzy sets theory (FST) and was introduced by Zadeh [86] in 1965 dealing with reasoning that is approximate rather than classical logic. A fuzzy variable is modeled by a membership function which assigns a degree of membership to a set. Usually, this degree of membership varies from zero to one. [87] presents a novel approach using approximate reasoning to determine suitable candidate nodes in a distribution system for capacitor placement. Voltages and power loss reduction of distribution system nodes are modelled by fuzzy membership functions. [88] developed a fuzzy-based approach for the placement of the shunt capacitor banks in a distribution system by considering harmonic distortions. In [89] voltage and real power loss index of distribution system nodes are modelled by fuzzy membership function. A fuzzy inference system containing a set of heuristic rules is designed to determine candidate nodes suitable for capacitor placement in the distribution system. Capacitors are placed on the nodes with highest sensitivity index. Masoum et.al. in [90] proposed a fuzzy-based approach for optimal placement and sizing of fixed capacitor banks in radial distribution networks in the presence of voltage and current harmonics. The objective function includes the cost of power losses, energy losses and capacitor banks. Using fuzzy set 55

17 theory, a suitable combination of objective function and constraints is generated as a criterion to select the most suitable bus for capacitor placement. The α-cut process is applied at each iteration to guarantee simultaneous improvements of objective function and satisfying given constraints. E. Particle Swarm Optimization (PSO) Particle swarm optimization (PSO) method is a population based evolutionary computation technique developed by Dr. Eberhart and Dr. Kennedy [91] in 1995, inspired by social behaviour of bird flocking or fish schooling. It utilizes a population of particles that fly through the problem hyperspace with given velocities. At each iteration, the velocities of the individual particles are stochastically adjusted according to the historical best position for the particle itself and the neighbouring best position. Both the particle best and the neighbourhood best are derived according to user defined fitness function [92]. The movement of each particle naturally evolves an optimal or near-optimal solution. A discrete version of PSO [93] is combined with a radial distribution power flow algorithm (RDPF) to form a hybrid PSO algorithm (HPSO). The former is employed as a global optimizer to find the global optimal solution while the later is used to calculate the objective function and to verify bus voltage limits. To include the presence of harmonics, the developed HPSO was integrated with a harmonic power flow algorithm (HPF). Tamer Mohamed et al. [94] presents a novel approach that determines the optimal location and size of capacitors on radial distribution systems to improve voltage profile and reduce the active power loss. Binary particle swarm optimization has been used for discrete optimization problem of optimal 56

18 capacitor placement in nonlinear loads.[95] highlights the PSO key features and advantages over other various optimization algorithms and also discusses PSO possible future applications in the area of electric power systems and its potential theoretical studies. Two new objective functions are defined in [96]. The first one is defined as the sum of reliability cost and investment cost and the second is defined by adding the reliability cost, cost of losses and investment cost. This problem is solved using a particle swarm optimization-based algorithm on a distribution network. [97] provides a comprehensive survey on the power system applications that have benefited from the powerful nature of PSO as an optimization technique. F. Ant Algorithm Ant algorithm has been inspired by the behaviour of real ant colonies. Real ants are capable of finding the shortest path from food sources to the nest without using visual cues and bring them back to their colony by the formation of unique trails. Therefore, through a collection of cooperative agents called ants, the near-optimal solution to capacitor placement problems can be effectively achieved. Bouri et al. [98] presents an ant colony approach for the optimization of shunt capacitor placement on distribution systems under capacitor switching constraints. In [99], a method employing the ant colony search algorithm (ACSA) is proposed to solve the capacitor placement problems. [100] aims to minimize the total active losses in electrical distribution systems by means of optimal capacitor bank placement. The proposed methodology to solve this optimization problem is the ant colony optimization (ACO). The gradient method is combined in order to accelerate the convergence of the ACO algorithm. 57